On 0-cycles with modulus

TL;DR

This paper establishes an exact sequence linking 0-cycles with modulus and relative K-theory on smooth surfaces, addressing questions about their relationship and limitations of localization sequences.

Contribution

It provides a new exact sequence connecting Chow groups with modulus and relative K-theory, and demonstrates the non-extension of localization sequences to Chow groups with modulus.

Findings

Established an exact sequence relating CH_0(X,D) and K_0(X,D)

Answered a question of Kerz and Saito for resolutions of singularities

Showed localization sequence does not extend to Chow groups with modulus

Abstract

Given a smooth surface $X$ over a field and an effective Cartier divisor $D$, we provide an exact sequence connecting $CH_0(X,D)$ and the relative $K$-group $K_0(X,D)$. We use this exact sequence to answer a question of Kerz and Saito whenever $X$ is a resolution of singularities of a normal surface. This exact sequence is used to show that the localization sequence for ordinary Chow groups does not extend to Chow groups with modulus.